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Abstract

We introduce a spectral-interferometry (SI) technique for measuring the complete intensity and phase of relatively long and very complex ultrashort pulses. Ordinarily, such a method would require a high-resolution spectrometer, but our method overcomes this need. It involves making multiple measurements using SI (in its SEA TADPOLE variation) at numerous delays, measuring many temporal pulselets within the pulse, and concatenating the resulting pulselets. Its spectral resolution is the inverse delay range—many times higher than that of the spectrometer used. Our simple proof-of-principle implementation of it provided 71 fs temporal resolution and a temporal range of 100 ps using a few-cm low-resolution spectrometer.

Figures (6)

Experimental setup for MUD TADPOLE. Both the unknown pulse and the reference pulse are coupled into two equal-length single-mode fibers. The unknown pulse passes through a delay stage, which provides the variable delay. Although here it is shown that the unknown pulse is delayed with respect to the reference, it is inconsequential which pulse is delayed with respect to the other. In the horizontal dimension, the light is collimated by the spherical lens and spectrally resolved by the spectrometer. In the vertical dimension, the beams cross at a slight angle resulting in spatial fringes at the camera.

The MUD TADPOLE retrieval algorithm using simulated data. In Step 1, each SEA TADPOLE trace—corresponding to a different temporal slice—is spatially Fourier filtered, resulting in the electric field at each delay, Ei(ω). In Step 2, the retrieved fields are temporally filtered, keeping only the region in which the unknown and reference pulses are temporally overlapped. Each retrieved field, Ei(ω), is Fourier transformed to the time domain and temporally shifted to the lab frame yielding E˜i,lab(t−τi). In the figure, each color represents the retrieved field at a different delay. Although, only the amplitudes are shown, after re-phasing, the same process is done with the retrieved phases. In Step 3, the retrieved amplitude and phase are separately concatenated using a weighted average, resulting in the retrieval of the entire unknown pulse.

Retrieving the phase and spectrum in SEA TADOLE. A typical SEA TADPOLE trace of a heavily chirped pulse is shown in 3(a). Fig 3(b) shows how the one-dimensional spatial Fourier transform separates the data into three bands. The side-bands contain both the spectral phase difference and the spectrum of the unknown pulse, and so one of these is kept, and then inverse Fourier transformed back to the position domain. Due to the fibers, no spatial information about the pulse is present, so we sum the resulting field over the camera’s position axis. At this point, the known reference pulse’s field is divided out in order to extract the unknown pulse’s intensity and phase.

a. The MUD TADPOLE-retrieved temporal amplitude and phase of a chirped 40 ps pulse. b. The MUD TADPOLE-retrieved spectrum compared to an independently measured spectrum using a spectrometer. c. The measured SEA TADPOLE spectrum compared to the independently measured spectrum. The overly narrow SEA TADPOLE spectrum shows the need for MUD TADPOLE. d. Concatenation of the retrieved temporal amplitudes, Ai(t-τi). Similar to Step 3 in Fig. 2, each color represents the retrieved amplitude at a different delay, which shows the multiple measurements overlapping in time as discussed in Section 3.3. e. Concatentation of the retrieved temporal phases after re-phasing, φi(t-τi).

A comparison of the measured and calculated temporal profiles of a chirped double pulse at variable delays. a,b. The MUD TADPOLE retrieved and simulated temporal profile of two 22 ps linearly chirped pulses separated by 1.6 ps. c,d. The retrieved and simulated temporal profile after increasing the delay between pulses to 4.6 ps. e,f. The retrieved and simulated temporal profile after increasing the delay between pulses to 9.2 ps. g,h. The retrieved and simulated temporal profile after increasing the delay between pulses to 24 ps. At this large delay the temporal phase develops a cusp which MUD TADPOLE is able to retrieve. i,j. The retrieved and simulated temporal profile of a 50 ps double pulse. At such a large delay the temporal beating is not as noticeable as at much shorter delays because fewer frequencies are temporally overlapped. In all examples the agreement between the retrieved and simulated results is good. These results simultaneously highlight the extended temporal range and high temporal resolution of MUD TADPOLE.

a. The MUD TADPOLE-retrieved temporal intensity and phase of a 50 ps chirped double pulse. b. The retrieved spectrum compared to an independently measured spectrum. The spectrum shows the high spectral resolution of MUD TADPOLE, and it highlights a conventional spectrometer’s limitions. c. Concatenation of the retrieved temporal amplitudes, Ai(t-τi). Similar to Step 3 in Fig. 2, each color represents the retrieved amplitude at a different delay. e. Concatentation of the retrieved temporal phases after re-phasing, φi(t-τi).